Despite being a key bottleneck in many machine learning tasks, the cost of solving large linear systems has proven challenging to quantify due to problem-dependent quantities such as condition numbers. To tackle this, we consider a fine-grained notion of complexity for solving linear systems, which is motivated by applications where the data exhibits low-dimensional structure, including spiked covariance models and kernel machines, and when the linear system is explicitly regularized, such as ridge regression. Concretely, let $\kappa_\ell$ be the ratio between the $\ell$th largest and the smallest singular value of $n\times n$ matrix $A$. We give a stochastic algorithm based on the Sketch-and-Project paradigm, that solves the linear system $Ax=b$ in time $\tilde O(\kappa_\ell\cdot n^2\log1/\epsilon)$ for any $\ell = O(n^{0.729})$. This is a direct improvement over preconditioned conjugate gradient, and it provides a stronger separation between stochastic linear solvers and algorithms accessing $A$ only through matrix-vector products. Our main technical contribution is the new analysis of the first and second moments of the random projection matrix that arises in Sketch-and-Project.

We study the interpolation capabilities of implicit neural representations (INRs) of images. In principle, INRs promise a number of advantages, such as continuous derivatives and arbitrary sampling, being freed from the restrictions of a raster grid. However, empirically, INRs have been observed to poorly interpolate between the pixels of the fit image; in other words, they do not inherently possess a suitable prior for natural images. In this paper, we propose to address and improve INRs’ interpolation capabilities by explicitly integrating image prior information into the INR architecture via deep decoder, a specific implementation of the deep image prior (DIP). Our method, which we call TITAN, leverages a residual connection from the input which enables integrating the principles of the grid-based DIP into the grid-free INR. Through super-resolution and computed tomography experiments, we demonstrate that our method significantly improves upon classic INRs, thanks to the induced natural image bias. We also find that by constraining the weights to be sparse, image quality and sharpness are enhanced, increasing the Lipschitz constant.

Seismic advances in generative AI algorithms for imagery, text, and other data types have led to the temptation to use AI-synthesized data to train next-generation models. Repeating this process creates an autophagous ("self-consuming") loop whose properties are poorly understood. We conduct a thorough analytical and empirical analysis using state-of-the-art generative image models of three families of autophagous loops that differ in how fixed or fresh real training data is available through the generations of training and whether the samples from previous-generation models have been biased to trade off data quality versus diversity. Our primary conclusion across all scenarios is that *without enough fresh real data in each generation of an autophagous loop, future generative models are doomed to have their quality (precision) or diversity (recall) progressively decrease.* We term this condition Model Autophagy Disorder (MAD), by analogy to mad cow disease, and show that appreciable MADness arises in just a few generations.

We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

Machine learning systems are often applied to data that is drawn from a different distribution than the training distribution. Recent work has shown that for a variety of classification and signal reconstruction problems, the out-of-distribution performance is strongly linearly correlated with the in-distribution performance. If this relationship or more generally a monotonic one holds, it has important consequences. For example, it allows to optimize performance on one distribution as a proxy for performance on the other. In this paper, we study conditions under which a monotonic relationship between the performances of a model on two distributions is expected. We prove an exact asymptotic linear relation for squared error and a monotonic relation for misclassification error for ridge-regularized general linear models under covariate shift, as well as an approximate linear relation for linear inverse problems.

Is overparameterization a privacy liability? In this work, we study the effect that the number of parameters has on a classifier’s vulnerability to membership inference attacks. We first demonstrate how the number of parameters of a model can induce a privacy-utility trade-off: increasing the number of parameters generally improves generalization performance at the expense of lower privacy. However, remarkably, we then show that if coupled with proper regularization, increasing the number of parameters of a model can actually simultaneously increase both its privacy and performance, thereby eliminating the privacy-utility trade-off. Theoretically, we demonstrate this curious phenomenon for logistic regression with ridge regularization in a bi-level feature ensemble setting. Pursuant to our theoretical exploration, we develop a novel leave-one-out analysis tool to precisely characterize the vulnerability of a linear classifier to the optimal membership inference attack. We empirically exhibit this “blessing of dimensionality” for neural networks on a variety of tasks using early stopping as the regularizer

Implicit neural representations (INRs) have recently advanced numerous vision-related areas. INR performance depends strongly on the choice of activation function employed in its MLP network. A wide range of nonlinearities have been explored, but, unfortunately, current INRs designed to have high accuracy also suffer from poor robustness (to signal noise, parameter variation, etc.). Inspired by harmonic analysis, we develop a new, highly accurate and robust INR that does not exhibit this trade off. Our Wavelet Implicit neural REpresentation (WIRE) uses as its activation function the complex Gabor wavelet that is well-known to be optimally concentrated in space-frequency and to have excellent biases for representing images. A wide range of experiments (image denoising, image inpainting, super-resolution, computed tomography reconstruction, image over fitting, and novel view synthesis with neural radiance fields) demonstrate that WIRE defines the new state of the art in INR accuracy, training time, and robustness.

High dimensionality poses many challenges to the use of data, from visualization and interpretation, to prediction and storage for historical preservation. Techniques abound to reduce the dimensionality of fixed-length sequences, yet these methods rarely generalize to variable-length sequences. To address this gap, we extend existing methods that rely on the use of kernels to variable-length sequences via use of the Recurrent Neural Tangent Kernel (RNTK). Since a deep neural network with ReLu activation is a Max-Affine Spline Operator (MASO), we dub our approach Max-Affine Spline Kernel (MASK). We demonstrate how MASK can be used to extend principal components analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) and apply these new algorithms to separate synthetic time series data sampled from second-order differential equations.

Among the most successful methods for sparsifying deep (neural) networks are those that adaptively mask the network weights throughout training. By examining this masking, or dropout, in the linear case, we uncover a duality between such adaptive methods and regularization through the so-called “η-trick” that casts both as iteratively reweighted optimizations. We show that any dropout strategy that adapts to the weights in a monotonic way corresponds to an effective subquadratic regularization penalty, and therefore leads to sparse solutions. We obtain the effective penalties for several popular sparsification strategies, which are remarkably similar to classical penalties commonly used in sparse optimization. Considering variational dropout as a case study, we demonstrate similar empirical behavior between the adaptive dropout method and classical methods on the task of deep network sparsification, validating our theory.

In this paper, we introduce a new online decision making paradigm that we call Thresholding Graph Bandits. The main goal is to efficiently identify a subset of arms in a multi-armed bandit problem whose means are above a specified threshold. While traditionally in such problems, the arms are assumed to be independent, in our paradigm we further suppose that we have access to the similarity between the arms in the form of a graph, allowing us to gain information about the arm means with fewer samples. Such a feature is particularly relevant in modern decision making problems, where rapid decisions need to be made in spite of the large number of options available. We present GrAPL, a novel algorithm for the thresholding graph bandit problem. We demonstrate theoretically that this algorithm is effective in taking advantage of the graph structure when the structure is reflective of the distribution of the rewards. We confirm these theoretical findings via experiments on both synthetic and real data.

Ensemble methods that average over a collection of independent predictors that are each limited to a subsampling of both the examples and features of the training data command a significant presence in machine learning, such as the ever-popular random forest, yet the nature of the subsampling effect, particularly of the features, is not well understood. We study the case of an ensemble of linear predictors, where each individual predictor is fit using ordinary least squares on a random submatrix of the data matrix. We show that, under standard Gaussianity assumptions, when the number of features selected for each predictor is optimally tuned, the asymptotic risk of a large ensemble is equal to the asymptotic ridge regression risk, which is known to be optimal among linear predictors in this setting. In addition to eliciting this implicit regularization that results from subsampling, we also connect this ensemble to the dropout technique used in training deep (neural) networks, another strategy that has been shown to have a ridge-like regularizing effect.

Algorithms often carry out equally many computations for "easy" and "hard" problem instances. In particular, algorithms for finding nearest neighbors typically have the same running time regardless of the particular problem instance. In this paper, we consider the approximate $k$-nearest-neighbor problem, which is the problem of finding a subset of O(k) points in a given set of points that contains the set of $k$ nearest neighbors of a given query point. We propose an algorithm based on adaptively estimating the distances, and show that it is essentially optimal out of algorithms that are only allowed to adaptively estimate distances. We then demonstrate both theoretically and experimentally that the algorithm can achieve significant speedups relative to the naive method.

Feature selection is an important challenge in machine learning. It plays a crucial role in the explainability of machine-driven decisions that are rapidly permeating throughout modern society. Unfortunately, the explosion in the size and dimensionality of real-world datasets poses a severe challenge to standard feature selection algorithms. Today, it is not uncommon for datasets to have billions of dimensions. At such scale, even storing the feature vector is impossible, causing most existing feature selection methods to fail. Workarounds like feature hashing, a standard approach to large-scale machine learning, helps with the computational feasibility, but at the cost of losing the interpretability of features. In this paper, we present MISSION, a novel framework for ultra large-scale feature selection that performs stochastic gradient descent while maintaining an efficient representation of the features in memory using a Count-Sketch data structure. MISSION retains the simplicity of feature hashing without sacrificing the interpretability of the features while using only O(log^2(p)) working memory. We demonstrate that MISSION accurately and efficiently performs feature selection on real-world, large-scale datasets with billions of dimensions.